Should I use Monte Carlo or a classifier for this Decision Making problem?
A classifier seems to me to not really fit the problem. I am not really sure. Actually, I don't see how a classifier like a decision tree, for example, would work here. Because I have to predict decision of the counselor to propose or not based on the decision of banks (and I don't know their criteria) to refuse or accept applicants who were proposed loan insurance and accepted it.
  
  The data I have is former applicants profile who were sent to banks and if they were accepted or not by the bank, if they wanted a loan insurance or not and the type of acquisition they wanted to make with their loan.


Why does this seem to you like something where a classifier wouldn't fit? Unless I'm missing something, it sounds like a prototypical example of a classification problem to me.

You have:


Input features (applicants' profile)
A clear (binary?) prediction target: propose or don't propose (equivalent to predicting whether or not the bank would accept, because you'll always want to propose if the bank would accept, and never propose if the bank wouldn't accept).
Old training data containing both the input features and the matching prediction targets.




Approaches like Monte-Carlo Tree Search can only be used if you have a forward model or simulator. In your setting, you could view the features (applicants' profile) as a "game state", and model the problem as a game with two actions (propose or not propose). However, you don't have a forward model (a function that, given a current state and action, generates a possible reward and subsequent state). 

In applications where MCTS is often used (such as games), you do have such a forward model: for a game like Go or chess, you can easily program the game's rules, program how you transition from one state into another when you select an action, etc. This does not appear to be the case for you.